Chapter 9Sequences and Series

The Fibonacci sequence is a series of integers mentioned in a book by Leonardo of Pisa (Fibonacci) in 1202 as the answer to an ancient arithmetic problem. The series begins with zero, and naturally progresses to one. The series becomes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on into infinity. In the Fibonacci sequence, each number is added to the previous to make the next.

9.1

Overview

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Example 1Example 2

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Sequences and Series

A series is the sum of the terms of a sequence. Finite sequences and series have defined first and

last terms Infinite sequences and series continue indefinitely.

9.2

Sequences

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Practice with examples: a) b) c) p.527

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Definitions:

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Examples:

Bounded and monotonic. Limit is 1

Bounded but non-monotonic.Limit does not exist

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Definitions: Arithmetic and Geometric sequences

An arithmetic sequence goes from one term to the next by adding (or subtracting) the same value.

Examples: 2, 5, 8, 11, 14,... (add 3 at each step)7, 3, –1, –5,... (subtract 4 at each step)

A geometric sequence goes from one term to the next by multiplying (or dividing) by the same value.

Examples: 1, 2, 4, 8, 16,... (multiply by 2 at each step)81, 27, 9, 3, 1, 1/3,... (divide by 3 at each step)

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Practice:

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Practice:

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Practice:

9.3

Infinite Series

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Recall:

A series is the sum of the terms of a sequence.

Finite sequences and series have defined first

and last terms.

Infinite sequences and series continue indefinitely.

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Geometric series Constant ratio between successive terms.

Example:

Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, and finance.

Common ratio: the ratio of successive terms in the series

Example:

The behavior of the terms depends on the common ratio r.

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If r is between −1 and +1, the terms of the series become smaller and smaller, and the series converges to a sum.

If r is greater than one or less than minus one the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, and the series has no sum. The series diverges.

If r is equal to one, all of the terms of the series are the same. The series diverges.

If r is minus one the terms take two values alternately (e.g. 2, −2, 2, −2, 2,... ). The sum of the terms oscillates between two values (e.g. 2, 0, 2, 0, 2,... ). This is a different type of divergence and again the series has no sum. (example Grandi's series: 1 − 1 + 1 − 1 + ···).

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Formula:

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Practice:

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Answers:

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Practice 2:

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Answers:

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Application of geometric series: Repeating decimals

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Telescoping series

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A term will cancel with a term that is farther down the list.

It’s not always obvious if a series is telescoping or not until you try to get the partial sums and then see if they are in fact telescoping.

9.4 – 9.5 – 9.6

Convergence Tests for Infinite Series

Alternating Series

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Let be an infinite series of positive terms.

The series converges if and only if the sequence of partial sums,

, converges. This means:

Definition of Convergence for an infinite series:

1nna

aaaaSn 321

1

limn

nnn

aS

If , the series diverges.

Divergence Test:0lim

nn

a

1nna

Example: The series

12 1n n

nis divergent since 1

11

1lim

1lim

22

nn

nnn

However, 0lim n

na does not imply convergence!

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Geometric Series:

The Geometric Series: converges for

If the series converges, the sum of the series is:

Example: The series with and converges .

The sum of the series is 35.

 

12 nararara 11 r

r

a

1

n

n

1 8

75

8

351 aa

8

7r

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p-Series:

The Series: (called a p-series) converges for

and diverges for

Example: The series is convergent. The series is divergent.

 

1p

1

1

npn

1p

1001.1

1

n n

1

1

n n

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Integral test:

If f is a continuous, positive, decreasing function on with

then the series converges if and only if the improper integral converges.

Example: Try the series: Note: in general for a series of the form:

13

1

n n

),1[ nanf )(

1nna

1

)( dxxf

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Comparison test:

If the series and are two series with positive terms, then:

(a) If is convergent and for all n, then converges.

(b) If is divergent and for all n, then diverges.

(smaller than convergent is convergent) (bigger than divergent is divergent)

Examples: which is a divergent harmonic series. Since the original series is larger by comparison, it is divergent.

which is a convergent p-series. Since the original

series is smaller by comparison, it is convergent.

1nna

1nnb

1nnb nn ba

1nna

1nnb nn ba

1nna

22

22

2

13

3

2

3

nnn nn

n

n

n

1

21

31

23

1

2

5

2

5

12

5

nnn nn

n

nn

n

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Limit Comparison test:

If the the series and are two series with positive terms, and if

where then either both series converge or both series diverge.

Useful trick: To obtain a series for comparison, omit lower order terms in the numerator and the denominator and then simplify.

Examples: For the series compare to which is a convergent p-series.

For the series compare to which is a divergent geometric

series.

1nna

1nnb c

b

a

n

n

n

lim

c0

12 3n nn

n

1 2

31

2

1

nn nn

n

123n

n

n

n

n

11 33 n

n

nn

n

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Alternating Series test:

If the alternating series satisfies:

and then the series converges.

Definition: Absolute convergence means that the series converges without alternating (all signs

and terms are positive).

Example: The series is convergent but not absolutely convergent.

Alternating p-series converges for p > 0.  

Example: The series and the Alternating Harmonic series are convergent.

1

65432111

nn

n bbbbbbb

1 nn bb 0lim n

nb

0 1

1

n

n

n

1

)1(

np

n

n

1

)1(

n

n

n

1

)1(

n

n

n

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Ratio test:

(a) If then the series converges;

(b) If the series diverges.

(c) Otherwise, you must use a different test for convergence.

If this limit is 1, the test is inconclusive and a different test is required.

Specifically, the Ratio Test does not work for p-series.

Example:

1lim 1

n

n

n a

a

1nna

1lim 1

n

n

n a

a

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Useful procedure:

Apply the following steps when testing for convergence:

1.Does the nth term approach zero as n approaches infinity? If not, the Divergence Test implies the series diverges.

2.Is the series one of the special types - geometric, telescoping, p-series, alternating series?

3.Can the integral test, ratio test, or root test be applied?

4.Can the series be compared in a useful way to one of the special types?

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Summaryof all tests: